Q K-TYPE SPACES OF QUATERNION-VALUED FUNCTIONS

In this paper we develop the necessary tools to generalize the QK -type function classes to the case of the monogenic functions defined in the unit ball of R3, some important basic properties of these functions are also considered. Further, we show some relations between QK(p, q) and α-Bloch spaces of quaternion-valued functions.

1.2.Quaternion function spaces.We will work throughout this paper in the field H (the skew field of quaternion-valued functions), i.e. each element a ∈ H with basis 1, e 1 , e 2 , e 3 , can be given in the form a := a 0 + a 1 e 1 + a 2 e 2 + a 3 e 3 , a k ∈ R, k = 0, 1, 2, 3 The multiplication rules of these elements are given by The conjugation element ā of an element a is ā = a 0 − a 1 e 1 − a 2 e 2 − a 3 e 3 , with the property x 2 e 2 be a quaternion point.
Given Ω ⊂ R 3 a domain and let f : Ω −→ H the quaternion-valued functions defined in Ω.For p ∈ N∪{0}, thus the notation C p (Ω; H) has the usual componentwise meaning.We consider D and D the generalization of a Cauchy-Riemann operator and it's conjugate, respectively, and they are defined on C 1 (Ω; H) by The equation Df = 0 has the solutions for all x ∈ Ω, are called left-hyperholomorphic functions and they are generalized of the analytic function classes from the functions in one complex variable theory.For more details about monogenic function classes and general Clifford analysis, we refer to [2,6,11] and others.
Let B be the unit ball in ⊂ R 3 , with boundary S = ∂B.The class M(B) consists of all monogenic functions on B. For r > 0 and a ∈ R 3 , let B(a, r) denote by the ball with center a and radius r.Also, for a ∈ B and 0 < R < 1, an Euclidean ball U (a, R) = {x : |ϕ a (x)| < R}, with center and radius, respectively, Let α > 0, the quaternion α-Bloch space B α (see [4,9]) defined by : If α = 3 2 , we have the standard quaternion Bloch space B. The space B α 0 is called the quaternion little α-Bloch, which consists of all f ∈ B α such that lim For f ∈ M(B), the weighted quaternion Dirichlet space D p,q , (0 < p < ∞, −2 < q < ∞), is given by: If q = 0, we have the space D 2,0 (the quaternion Dirichlet space D).
For 0 < p < ∞, −1 < q < ∞, define the D K (p, q) quaternion Dirichlet-type space as the set of f ∈ M(B) satisfying From the definition of Q K (p, q) spaces the following lemma become immediate with a = 0.
From now, we assume that Otherwise, Q K (p, q) contains only constant functions.

Fact 1
Let 0 ≤ p < ∞, −1 < q < ∞, and let f ∈ M(B) be a non-constant function.If ( 1.1) does not hold, then Proof.Let f ∈ Q K (p, q) be a non constant function.Then, there is x 0 ∈ B and 0 < R < 1 such that |Df (x)| > 0 for each x ∈ B(x 0 , R).Thus by Lemma 1.1 and subharmonicity of |Df | p where A(x 0 , R) = where dσ denotes the normalized surface element in S.This is a contradiction; therefor f is constant and the fact is proved.
In this work, we introduce a classes of H-valued functions on R 3 .These classes are so called Q K (p, q) spaces of monogenic function.We will study these classes and their relations to the quaternion α-Bloch space.We shall prove some basic properties concerning Q K (p, q) and Q K,0 (p, q) spaces in hyperholomorphic functions.Our results in this work are extensions of our results in [1] and the results due to Essén and Wulan (see [3]) in hyperholomorphic functions case.For simplicity we restricted us toR 3 the lowest noncommutative case and quaternion-valued functions.Next, the hyperholomorphic function spaces were the aim of many works as [1,4,8] and [9].
In particular, we will need the following results for quaternion sense in the sequel: (1.2) Then, for every a ∈ B, we have where C = 48 π .
Remark 1.2.The problem in quaternion sense is that, Df (x) is monogenic, but Df (ϕ a (w)) is not monogenic.From [10] we know that 1− wa |1−āw| 3 Df (ϕ a (w)) is hyperholomorphic.So, by the Jacobian determinant , which has no singularities we can solve this problem.

Characterizations of Q K (p, q) classes
In this part, we prove some essential properties of quaternion Q K (p, q) spaces as basic scale properties.
Proposition 2.1.Let K satisfy (1.1) and let f ∈ M(B), 1 ≤ p < ∞, and −2 < q < ∞.Then, we have Proof.Since 0 < R < 1, by Lemma 1.4 after the change of variable x = ϕ a (w) then we deduce that Proof.Since 1 ≤ p < ∞, t is easy to prove that .K is a norm.To show the completeness of ( .K , Q K (p, q)), fix 0 < R < 1. Applying Proposition 2.1, we obtain By the fact that (1 − |x| 2 ) 3 ≈ |U (a, R)|, from Lemma 1.2 and Lemma 1.3, we get , where a positive constant C 1 (R, p, q) is depending on R, p and q, which implies that where the constant C = max 1, C(M, C 1 (R, p, q), q+3 p ) .Now, we let {f n } be a Cauchy sequence in Q K (p, q) spaces.From (2.1) we deduce that {f n } is also a Cauchy sequence in the topology of uniform convergence on compact sets.Thus thereis a function 2 for all n, k ∈≥ N.For each a ∈ B and n ≥ N, by applying Fatou's lemma, we obtain Thus, for all n ≥ N, which implies that f n → f in Q K (p, q).Hence, the norm .K is complete, therefore Q K (p, q) spaces is a Banach space in Clifford setting.
3. The Quaternion Bloch and Q K (p, q) Spaces In this part of the paper, we consider the relations between Q K (p, q) and α-Bloch spaces in quaternion sense.We characterize the quaternion α-Bloch spaces by the help of integral norms of quaternion Q K (p, q) spaces.Our results extend the results due to Wulan and Zhou [12] in quaternion sense.
Proof.(i) Let 0 < R < 1 be fixed and a ∈ B. From Proposition 2.1, we acquire Now we change the variable x = ϕ a (w), then we acquire Combining Theorem 3.1, we deduce the following corollary: if and only if there is an p , and a ∈ B. Then for any 0 < R < 1, we deduce Conversely, let (3.2) holds then, we deduce

Conclusion.
Our results in this work will be of important uses in the study of operator theory at the interface of monogenic function spaces.This work is a try to synthesize the achievements in the properties of monogenic Q K (p, q) function spaces.The problem in quaternion sense is that, Df (x) is monogenic, but Df (φ(x)) is not monogenic, where φ : B → B is a monogenic function.The following question is open problem: What properties of operators act between this classes of monogenic functions, like F (p, q, s) and Q K (p, q) classes?
In quaternion case, several authors have studied function spaces and classes like Q p , Q K classes and F (p, q, s) spaces, see [1,3,8] and others.

1 . Introduction 1 . 1 .
Analytic function spaces.The so called Q K -type spaces of analytic functions on D = {z ∈ C : |z| <
q), then by estimate above we have f ∈ B q+3 p .(ii) Let f ∈ B q+3 p be non constant.Then, there is M > 0 constant such that (1 − |a| 2 ) q+3 p |Df (a)| ≤ M, for all x ∈ B.